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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
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Thank you @Carcass, I'll follow the rules next time
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
MyGuruSteve wrote:
Farina wrote:
If |0.1x - 3| >= 1, then x could be which of the following values?
Indicate all such values.

A.10
B.20
C.30
D.40
E.50
F.60

PS: I am sorry I dont know how to add options


You can either backsolve from the answer choices for this question, or solve it algebraically:

Working backwards from the answers:

If \( x = 30\), then \(0.1x = 3\), so \(|0.1(30) - 3|=0\), which doesn't fit. For choices D, E, and F, the number inside the absolute value markers is already a positive number >= 1, so those are valid answer choices. For choices A and B, the number inside the absolute value markers would be -2 and -1 respectively, the absolute values of which are 2 and 1. All answer choices are valid except for C.

Algebraic Method


Remember that inequalities containing absolute values can be written as two separate inequalities:

\(|0.1x-3|>=1\)

\(0.1x-3>=1\) or \(0.1x-3 <= -1\)

\(0.1x>=4\) or \(0.1x <= 2\)

\(x>=40\) or \(x<= 20\)

The only choice that x cannot be is 30


Answer: A,B,D,E,F


Hi thanks for your reply. Just one confusion about flipping of sign here.
0.1x - 3 >= -1
0.1x >= -1+3
0.1x >= 2

why the sign will flip when there is no negative sign?
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
Expert Reply
Farina wrote:
MyGuruSteve wrote:
Farina wrote:
If |0.1x - 3| >= 1, then x could be which of the following values?
Indicate all such values.

A.10
B.20
C.30
D.40
E.50
F.60

PS: I am sorry I dont know how to add options


You can either backsolve from the answer choices for this question, or solve it algebraically:

Working backwards from the answers:

If \( x = 30\), then \(0.1x = 3\), so \(|0.1(30) - 3|=0\), which doesn't fit. For choices D, E, and F, the number inside the absolute value markers is already a positive number >= 1, so those are valid answer choices. For choices A and B, the number inside the absolute value markers would be -2 and -1 respectively, the absolute values of which are 2 and 1. All answer choices are valid except for C.

Algebraic Method


Remember that inequalities containing absolute values can be written as two separate inequalities:

\(|0.1x-3|>=1\)

\(0.1x-3>=1\) or \(0.1x-3 <= -1\)

\(0.1x>=4\) or \(0.1x <= 2\)

\(x>=40\) or \(x<= 20\)

The only choice that x cannot be is 30


Answer: A,B,D,E,F


Hi thanks for your reply. Just one confusion about flipping of sign here.
0.1x - 3 >= -1
0.1x >= -1+3
0.1x >= 2

why the sign will flip when there is no negative sign?


Think of "Absolute value" as "distance away from zero on a number line". No matter whether you're going in a positive or a negative direction, the actual distance is always going to be positive (asan analogy, think about going 50 miles east or 50 miles west on a map---no matter which direction you go, you've still gone 50 miles.)

The inequality \(|0.1x-3|>=1\) tells you that whatever is inside the absolute value sign "||" is 1 or more units away from zero. We can see that that's obviously true for numbers like 1,2,2.5,3, etc....but it's also true for -1, -2, -2.5, -3, etc.

So whenever you have an absolute value inequality, you'll typically need to set up two equations, one for the positive direction, and one for the negative. Then solve each separately, to find the range of possible values.
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
Quote:
Think of "Absolute value" as "distance away from zero on a number line". No matter whether you're going in a positive or a negative direction, the actual distance is always going to be positive (asan analogy, think about going 50 miles east or 50 miles west on a map---no matter which direction you go, you've still gone 50 miles.)

The inequality \(|0.1x-3|>=1\) tells you that whatever is inside the absolute value sign "||" is 1 or more units away from zero. We can see that that's obviously true for numbers like 1,2,2.5,3, etc....but it's also true for -1, -2, -2.5, -3, etc.

So whenever you have an absolute value inequality, you'll typically need to set up two equations, one for the positive direction, and one for the negative. Then solve each separately, to find the range of possible values.


You mean to say whenever we set up the equation for negative value we have to flip the sign from start? and no need to wait for the answer i.e if answer is negative only then flip the sign?
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
Expert Reply
Farina wrote:
Quote:
Think of "Absolute value" as "distance away from zero on a number line". No matter whether you're going in a positive or a negative direction, the actual distance is always going to be positive (asan analogy, think about going 50 miles east or 50 miles west on a map---no matter which direction you go, you've still gone 50 miles.)

The inequality \(|0.1x-3|>=1\) tells you that whatever is inside the absolute value sign "||" is 1 or more units away from zero. We can see that that's obviously true for numbers like 1,2,2.5,3, etc....but it's also true for -1, -2, -2.5, -3, etc.

So whenever you have an absolute value inequality, you'll typically need to set up two equations, one for the positive direction, and one for the negative. Then solve each separately, to find the range of possible values.


You mean to say whenever we set up the equation for negative value we have to flip the sign from start? and no need to wait for the answer i.e if answer is negative only then flip the sign?


The purpose of the two equations (or inequalities) is to solve for the values both above your starting point (i.e. in a positive direction) and below (in a negative direction).

Say you had something like
|x| > 3

This is really saying that "the distance away from zero is greater than 3 units.

So the first equation would simply be x > 3.

But the numbers less than -3 are ALSO more than 3 units away from zero, just in the opposite direction.

In other words, it's also true that x < -3.

So, for inequalities with absolute values, you'll have two equations:

Equation #1 -- simply remove the || sign.
Equation #2-- make the right side negative, and flip the sign.

Here's an example:

|x+2| > 4

First Equation (simply remove the || sign):
x+2 > 4

Second Equation:
x+2 < -4
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
1
MyGuruSteve wrote:
Farina wrote:
Quote:
Think of "Absolute value" as "distance away from zero on a number line". No matter whether you're going in a positive or a negative direction, the actual distance is always going to be positive (asan analogy, think about going 50 miles east or 50 miles west on a map---no matter which direction you go, you've still gone 50 miles.)

The inequality \(|0.1x-3|>=1\) tells you that whatever is inside the absolute value sign "||" is 1 or more units away from zero. We can see that that's obviously true for numbers like 1,2,2.5,3, etc....but it's also true for -1, -2, -2.5, -3, etc.

So whenever you have an absolute value inequality, you'll typically need to set up two equations, one for the positive direction, and one for the negative. Then solve each separately, to find the range of possible values.


You mean to say whenever we set up the equation for negative value we have to flip the sign from start? and no need to wait for the answer i.e if answer is negative only then flip the sign?


The purpose of the two equations (or inequalities) is to solve for the values both above your starting point (i.e. in a positive direction) and below (in a negative direction).

Say you had something like
|x| > 3

This is really saying that "the distance away from zero is greater than 3 units.

So the first equation would simply be x > 3.

But the numbers less than -3 are ALSO more than 3 units away from zero, just in the opposite direction.

In other words, it's also true that x < -3.

So, for inequalities with absolute values, you'll have two equations:

Equation #1 -- simply remove the || sign.
Equation #2-- make the right side negative, and flip the sign.

Here's an example:

|x+2| > 4

First Equation (simply remove the || sign):
x+2 > 4

Second Equation:
x+2 < -4


Thank you very much for your help. It was a very important point. I used to get confused a lot in inequalities. Hope I'll do better now
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Re: If |0.1x - 3| >= 1, then x could be which of the following [#permalink]
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